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Transactions of the American Mathematical Society

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An enhancement of Zagier's polylogarithm conjecture


Author: Nobuo Sato
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 11G55, 11M32, 11R42
DOI: https://doi.org/10.1090/tran/7629
Published electronically: May 9, 2019
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Abstract: Let $ m\geq 2$ be a natural number and let $ \mathcal {A}$ be an ideal class of an imaginary quadratic number field. Zagier and Gangl constructed $ \mathbb{C}/\mathbb{Q}(m)$-valued invariants $ I_{m}(\mathcal {A})$ which they named ``the enhanced zeta value'', since the real part of $ i^{m-1}I_{m}(\mathcal {A})$, after being multiplied by a certain elementary factor in terms of a factorial and a power of $ 2\pi $, equals the partial zeta value $ \zeta (m,\mathcal {A})$. They also constructed the enhanced polylogarithm, a $ \mathbb{C}/\mathbb{Q}(m)$-valued function on the $ m$-th Bloch group $ \mathcal {B}_{m}(\mathbb{C})$, and formulated an enhanced conjecture for $ I_{m}(\mathcal {A})$ that gives a natural lift of the polylogarithm conjecture for $ \zeta (m,\mathcal {A})$ to a conjectural equality in $ \mathbb{C}/\mathbb{Q}(m)$. In this article, we define the Shintani L-function of two variables which is naturally regarded as a two-variable analog of the partial zeta function for imaginary quadratic fields. Then we study its analytic properties in order to construct $ \mathbb{C}/\mathbb{Q}(1)$-valued invariants $ \Lambda _{i}(1-m,\mathcal {A})$ ( $ i\in \{1,2\}$) for a ray class $ \mathcal {A}$ using the first partial derivative of the Shintani L-function at $ (1-m,1-m)$. From the construction, $ \Lambda _{1}(1-m,\mathcal {A})$ and $ \Lambda _{2}(1-m,\mathcal {A})$ are complex conjugate invariants that satisfy $ \zeta '(1-m,\mathcal {A})=\Lambda _{1}(1-m,\mathcal {A})+\Lambda _{2}(1-m,\mathcal {A})$. Then we prove the main theorem of this article about the equality between Zagier and Gangl's enhanced zeta value $ I_{m}(\mathcal {A})$ and $ \Lambda _{1}(1-m,\mathcal {A})$, by explicit calculation of the Fourier expansion of the partial derivative of the Shintani L-function. Finally, we formulate the enhanced conjecture for the ray class invariants $ \Lambda _{i}(1-m,\mathcal {A})$, by which we expand Zagier-Gangl's original conjecture. We also give several numerical examples to verify the correctness of our enhanced conjecture.


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Additional Information

Nobuo Sato
Affiliation: National Center for Theoretical Sciences, National Taiwan University, No. 1, Sec. 4, Roosevelt Road, Taipei, Taiwan
Address at time of publication: Kyushu University, 744 Motooka Nishi-ku, Fukuoka 819-0395
Email: saton@ncts.ntu.edu.tw; n-sato@math.kyushu-u.ac.jp

DOI: https://doi.org/10.1090/tran/7629
Received by editor(s): November 22, 2017
Received by editor(s) in revised form: June 1, 2018
Published electronically: May 9, 2019
Article copyright: © Copyright 2019 American Mathematical Society